Quantum algebras are normally considered together with a distinguished collection (`quantum group') of their symmetries that form a sort of quantum dynamical system (a precise definition is given below). A good example in this context is the quantum plane \(\mathbb {C}_q[x,y]\) , which is among the simplest quantum algebras. It is very well known that \(\mathbb {C}_q[x,y]\)  carries a \(U_q(\mathfrak {sl}_2)\) -symmetry (in other terms, a structure of \(U_q(\mathfrak {sl}_2)\) -module algebra, see, e.g., [1]}). During a long time it was the only such symmetry considered in the literature.
